{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "import sympy as sy\n",
    "sy.init_printing() "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <font face=\"gotham\" color=\"purple\"> Null Space </font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The **null space**, denoted as $\\text{Nul}A$ is the solution set of a homogeneous linear system, i.e. $Ax=0$. \n",
    "\n",
    "A null space is a always a subspace of $\\mathbb{R}^n$, why? Because the solution can always be at origin $(0, 0, ...)$.\n",
    "\n",
    "As an example, consider a linear system.\n",
    "\n",
    "$$\n",
    "2x_1-x_2+x_3 = 0\\\\\n",
    "x_1+2x_2+3x_3= 0 \n",
    "$$\n",
    "\n",
    "The augmented matrix is \n",
    "\n",
    "$$\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "2 & -1 & 1 & 0\\\\\n",
    "1 & 2 & 3 & 0\n",
    "\\end{matrix}\n",
    "\\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Before solving the system, we have already known there is no unique solution since a free variable presents, due to fact that two equation with three variables.\n",
    "\n",
    "Solve for the reduced echelon form."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & 0 & 1 & 0\\\\0 & 1 & 1 & 0\\end{matrix}\\right], \\  \\left( 0, \\  1\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  0  1  0⎤        ⎞\n",
       "⎜⎢          ⎥, (0, 1)⎟\n",
       "⎝⎣0  1  1  0⎦        ⎠"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Aug = sy.Matrix([[2,-1,1,0],[1,2,3,0]])\n",
    "Aug.rref()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$x_3$ is a free variable, the solution set can be written as\n",
    "\n",
    "$$\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "x_1 \\\\ x_2 \\\\ x_3\n",
    "\\end{matrix}\n",
    "\\right]=\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "-x_3 \\\\ -x_3 \\\\ x_3\n",
    "\\end{matrix}\n",
    "\\right]=\n",
    "x_3\\left[\n",
    "\\begin{matrix}\n",
    "-1 \\\\ -1 \\\\ 1\n",
    "\\end{matrix}\n",
    "\\right]\n",
    "$$\n",
    "\n",
    "which is a line passing both origin $(0, 0, 0)$ and $(-1, -1, 1)$, also a subspace of $\\mathbb{R}^3$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider another example, suppose we have an augmented matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-3 & 6 & -1 & 1 & -7 & 0\\\\1 & -2 & 2 & 3 & -1 & 0\\\\2 & -4 & 5 & 8 & -4 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-3  6   -1  1  -7  0⎤\n",
       "⎢                    ⎥\n",
       "⎢1   -2  2   3  -1  0⎥\n",
       "⎢                    ⎥\n",
       "⎣2   -4  5   8  -4  0⎦"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Aug = sy.Matrix([[-3,6,-1,1,-7,0],[1,-2,2,3,-1,0],[2,-4,5,8,-4,0]]);Aug"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & -2 & 0 & -1 & 3 & 0\\\\0 & 0 & 1 & 2 & -2 & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\end{matrix}\\right], \\  \\left( 0, \\  2\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  -2  0  -1  3   0⎤        ⎞\n",
       "⎜⎢                   ⎥        ⎟\n",
       "⎜⎢0  0   1  2   -2  0⎥, (0, 2)⎟\n",
       "⎜⎢                   ⎥        ⎟\n",
       "⎝⎣0  0   0  0   0   0⎦        ⎠"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Aug.rref()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The solution can be written as:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "x_1 \\\\ x_2 \\\\ x_3 \\\\x_4 \\\\ x_5\n",
    "\\end{matrix}\n",
    "\\right]=\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "2x_2+x_4-3x_5 \\\\ x_2 \\\\ -2x_4+2x_5 \\\\x_4 \\\\ x_5\n",
    "\\end{matrix}\n",
    "\\right]=\n",
    "x_2\\left[\n",
    "\\begin{matrix}\n",
    "2 \\\\ 1 \\\\ 0 \\\\0 \\\\ 0\n",
    "\\end{matrix}\n",
    "\\right]\n",
    "+\n",
    "x_4\\left[\n",
    "\\begin{matrix}\n",
    "1 \\\\ 0 \\\\ -2 \\\\1 \\\\ 0\n",
    "\\end{matrix}\n",
    "\\right]\n",
    "+x_5\\left[\n",
    "\\begin{matrix}\n",
    "-3 \\\\ 0 \\\\ 2 \\\\0 \\\\ 1\n",
    "\\end{matrix}\n",
    "\\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The $\\text{Nul}A$ is a subspace in $\\mathbb{R}^5$ with $\\text{dim}A=3$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <font face=\"gotham\" color=\"purple\"> Null Space vs Col Space </font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider matrix $A$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2 & 4 & -2 & 1\\\\-2 & -5 & 7 & 3\\\\3 & 7 & -8 & 6\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡2   4   -2  1⎤\n",
       "⎢             ⎥\n",
       "⎢-2  -5  7   3⎥\n",
       "⎢             ⎥\n",
       "⎣3   7   -8  6⎦"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = sy.Matrix([[2,4,-2,1],[-2,-5,7,3],[3,7,-8,6]]);A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Column space is a subspace in $\\mathbb{R}^n$, what is $n$? It is the number of rows, $n=3$.\n",
    "\n",
    "Null space is a subspace in $\\mathbb{R}^m$, what is $m$? It is the number of columns, $m=4$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How to find any nonzero vector in $\\text{Col}A$ and in $\\text{Nul}A$?\n",
    "\n",
    "Any column in a matrix can be a nonzero vector in $\\text{Col}A$, for instance first column: $(2, -2, 3)^T$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But to find a nonzero vector in null space requires some effort, construct the augmented matrix then turn it into rref."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & 0 & 9 & 0 & 0\\\\0 & 1 & -5 & 0 & 0\\\\0 & 0 & 0 & 1 & 0\\end{matrix}\\right], \\  \\left( 0, \\  1, \\  3\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  0  9   0  0⎤           ⎞\n",
       "⎜⎢              ⎥           ⎟\n",
       "⎜⎢0  1  -5  0  0⎥, (0, 1, 3)⎟\n",
       "⎜⎢              ⎥           ⎟\n",
       "⎝⎣0  0  0   1  0⎦           ⎠"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Aug = sy.Matrix([[2,4,-2,1,0],[-2,-5,7,3,0],[3,7,-8,6,0]]);Aug.rref()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The solution set with a free variable $x_3$ (because column 3 has no pivot) is \n",
    "\n",
    "$$\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "x_1 \\\\ x_2 \\\\ x_3\\\\x_4\n",
    "\\end{matrix}\n",
    "\\right]=\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "-9x_3 \\\\ 5x_3 \\\\ x_3\\\\0\n",
    "\\end{matrix}\n",
    "\\right]\n",
    "$$\n",
    "\n",
    "If we pick $x_3 =1$, a nonzero vector in $\\text{Nul}A$ is $(-9, 5, 1, 0)^T$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now consider two vectors\n",
    "\n",
    "$$\n",
    "u = \\left[\n",
    "\\begin{matrix}\n",
    "3 \\\\ -2 \\\\ -1\\\\ 0 \n",
    "\\end{matrix}\n",
    "\\right],\\qquad\n",
    "v = \\left[\n",
    "\\begin{matrix}\n",
    "3 \\\\ -1\\\\3\n",
    "\\end{matrix}\n",
    "\\right]\\\\\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Is $u$ in $\\text{Nul}A$? It can be verified easily"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0\\\\-3\\\\3\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡0 ⎤\n",
       "⎢  ⎥\n",
       "⎢-3⎥\n",
       "⎢  ⎥\n",
       "⎣3 ⎦"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u = sy.Matrix([[3],[-2],[-1],[0]])\n",
    "A*u"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$Au\\neq \\mathbf{0}$, therefore $u$ is not in $\\text{Nul}A$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Is $v$ in $\\text{Col}A$? Construct an augmented matrix with $v$, then solve it"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & 0 & 9 & 0 & 5\\\\0 & 1 & -5 & 0 & - \\frac{30}{17}\\\\0 & 0 & 0 & 1 & \\frac{1}{17}\\end{matrix}\\right], \\  \\left( 0, \\  1, \\  3\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  0  9   0   5  ⎤           ⎞\n",
       "⎜⎢                 ⎥           ⎟\n",
       "⎜⎢             -30 ⎥           ⎟\n",
       "⎜⎢0  1  -5  0  ────⎥, (0, 1, 3)⎟\n",
       "⎜⎢              17 ⎥           ⎟\n",
       "⎜⎢                 ⎥           ⎟\n",
       "⎝⎣0  0  0   1  1/17⎦           ⎠"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = sy.Matrix([[3],[-1],[3]])\n",
    "A.row_join(v).rref()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The augmented matrix show there are solutions, i.e. $v$ is a linear combination of its column space basis, so $v$ is in $\\text{Col}A$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <font face=\"gotham\" color=\"purple\"> Row Space </font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The **Row space** denoted as $\\text{Row}A$, contains all linear combination of row vectors and subspace in $\\mathbb{R}^n$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we perform row operations on $A$ to obtain $B$, both matrices have the same row space, because $B$'s rows are linear combinations of $A$'s. However, row operation will change the row dependence. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <font face=\"gotham\" color=\"purple\"> An Example </font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Find the row, column and null space of "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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9LnmZ3uJ7sH497NoQaAIB6eCNGD3bxCwKy97Z5QvrXsJq9tX2+LJLQ2BfEdj6aeffu73pQpLoTKsAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-2 & -5 & 8 & 0 & -17\\\\1 & 3 & -5 & 1 & 5\\\\3 & 11 & -19 & 7 & 1\\\\1 & 7 & -13 & 5 & -3\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-2  -5   8   0  -17⎤\n",
       "⎢                   ⎥\n",
       "⎢1   3   -5   1   5 ⎥\n",
       "⎢                   ⎥\n",
       "⎢3   11  -19  7   1 ⎥\n",
       "⎢                   ⎥\n",
       "⎣1   7   -13  5  -3 ⎦"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = sy.Matrix([[-2, -5, 8, 0, -17],\n",
    "               [1, 3, -5, 1, 5], \n",
    "               [3, 11, -19, 7, 1], \n",
    "               [1, 7, -13, 5, -3]]);A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & 0 & 1 & 0 & 1\\\\0 & 1 & -2 & 0 & 3\\\\0 & 0 & 0 & 1 & -5\\\\0 & 0 & 0 & 0 & 0\\end{matrix}\\right], \\  \\left( 0, \\  1, \\  3\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  0  1   0  1 ⎤           ⎞\n",
       "⎜⎢               ⎥           ⎟\n",
       "⎜⎢0  1  -2  0  3 ⎥           ⎟\n",
       "⎜⎢               ⎥, (0, 1, 3)⎟\n",
       "⎜⎢0  0  0   1  -5⎥           ⎟\n",
       "⎜⎢               ⎥           ⎟\n",
       "⎝⎣0  0  0   0  0 ⎦           ⎠"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = A.rref();B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The basis of the row space of $B$ is its first 3 rows: $(1,0,1,0,1), (0, 1, -2, 0, 3), (0, 0, 0, 1, -5)$ which are also the basis of the row space of $A$. However it does not necessarily mean that first 3 rows of $A$ forms the basis for row space, because the dependence among rows changed by row operation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In constrast, the basis of col space of $A$ is $(-2, 1, 3, 1)^T, (-5, 3, 11, 7)^T, (0, 1, 7, 5)^T$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & 0 & 1 & 0 & 1 & 0\\\\0 & 1 & -2 & 0 & 3 & 0\\\\0 & 0 & 0 & 1 & -5 & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\end{matrix}\\right], \\  \\left( 0, \\  1, \\  3\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  0  1   0  1   0⎤           ⎞\n",
       "⎜⎢                  ⎥           ⎟\n",
       "⎜⎢0  1  -2  0  3   0⎥           ⎟\n",
       "⎜⎢                  ⎥, (0, 1, 3)⎟\n",
       "⎜⎢0  0  0   1  -5  0⎥           ⎟\n",
       "⎜⎢                  ⎥           ⎟\n",
       "⎝⎣0  0  0   0  0   0⎦           ⎠"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Aug = A.row_join(sy.zeros(4,1));Aug.rref()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The null space is \n",
    "\n",
    "$$\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "x_1 \\\\ x_2 \\\\ x_3\\\\x_4 \\\\x_5\n",
    "\\end{matrix}\n",
    "\\right]=\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "-x_3-x_5 \\\\ 2x_3-3x_5 \\\\ x_3\\\\5x_5 \\\\x_5\n",
    "\\end{matrix}\n",
    "\\right]=\n",
    "x_3\\left[\n",
    "\\begin{matrix}\n",
    "-1 \\\\ 2 \\\\ 1\\\\0 \\\\0\n",
    "\\end{matrix}\n",
    "\\right]+\n",
    "x_5\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "-1 \\\\ -3 \\\\ 0\\\\5 \\\\1\n",
    "\\end{matrix}\n",
    "\\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <font face=\"gotham\" color=\"purple\"> Rank </font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Definition of rank:\n",
    "The **rank** is the dimension of the column space of $A$. The **nullity** of $A$ is the dimension of the null space."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <font face=\"gotham\" color=\"purple\"> The Rank Theorem</font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The dimensions of the column space and the row space of an $m \\times n$ matrix $A$ are equal that is why we only need to say rank is the dimension of the column space. \n",
    "\n",
    "This common dimension, the rank of $A$, also equals the number of pivot positions in $A$ and satisfies the equation\n",
    "$$\n",
    "\\operatorname{rank} A+\\operatorname{dim} \\mathrm{Nul} A=n\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The intuition is that when a matrix $A$ is converted into rref $B$, we can indirectly(matching the same column from $B$ to $A$) see the basis of column space, those columns in corresponding rref have pivots. \n",
    "\n",
    "And in rref, we can also see the basis of row space directly, every row in the basis of row space must have a pivot. And those rows which does not have pivots are for free variables, which is the dimension of null space. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <font face=\"gotham\" color=\"purple\"> Example 1 </font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If $A$ is $45 \\times 50$ matrix with a $10$-dimension nullity, what is the rank of $A$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$10$-$D$ nullity means 10 free variables, so the pivots are $50-10=40$, which is also the rank of $A$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <font face=\"gotham\" color=\"purple\"> Example 2 </font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The matrices below are row equivalent.\n",
    "$$\n",
    "A=\\left[\\begin{array}{rrrrr}\n",
    "2 & -1 & 1 & -6 & 8 \\\\\n",
    "1 & -2 & -4 & 3 & -2 \\\\\n",
    "-7 & 8 & 10 & 3 & -10 \\\\\n",
    "4 & -5 & -7 & 0 & 4\n",
    "\\end{array}\\right], \\quad B=\\left[\\begin{array}{rrrrr}\n",
    "1 & -2 & -4 & 3 & -2 \\\\\n",
    "0 & 3 & 9 & -12 & 12 \\\\\n",
    "0 & 0 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 0 & 0\n",
    "\\end{array}\\right]\n",
    "$$\n",
    "1. Find rank $A$ and $\\operatorname{dim}$ Nul $A$\n",
    "2. Find bases for Col $A$ and Row $A$.\n",
    "3. What is the next step to perform to find a basis for Nul $A$ ?\n",
    "4. How many pivot columns are in a row echelon form of $A^{T} ?$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. $rank(A)=2$, because $B$ has two pivots. And nullity is the number of free variables, there are 3, so $\\text{dim Nul}A = 3$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & 0 & 2 & -5 & 6 & 0\\\\0 & 1 & 3 & -4 & 4 & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\end{matrix}\\right], \\  \\left( 0, \\  1\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  0  2  -5  6  0⎤        ⎞\n",
       "⎜⎢                 ⎥        ⎟\n",
       "⎜⎢0  1  3  -4  4  0⎥        ⎟\n",
       "⎜⎢                 ⎥, (0, 1)⎟\n",
       "⎜⎢0  0  0  0   0  0⎥        ⎟\n",
       "⎜⎢                 ⎥        ⎟\n",
       "⎝⎣0  0  0  0   0  0⎦        ⎠"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = sy.Matrix([[2,-1,1,-6,8,0],\n",
    "               [1,-2,-4,3,-2,0],\n",
    "               [-7,8,10,3,-10,0],\n",
    "               [4,-5,-7,0,4,0]])\n",
    "A.rref()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. Bases for $\\text{Col}A$ is $(2,1,-7,4)^T, (-1,-2,8,-5)^T$, and for $\\text{Row}A$ is $(1,-2,-4,3,-2),(0,3,9,-12,12)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The $\\text{Nul}A$ and basis is\n",
    "\n",
    "$$\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "x_1 \\\\ x_2 \\\\ x_3\\\\x_4 \\\\x_5\n",
    "\\end{matrix}\n",
    "\\right]=\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "-2x_3+5x_4-6x_5 \\\\ -3x_3+4x_4-4x_5 \\\\ x_3\\\\x_4 \\\\x_5\n",
    "\\end{matrix}\n",
    "\\right]=\n",
    "x_3\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "-2 \\\\ -3 \\\\ 1\\\\0 \\\\0\n",
    "\\end{matrix}\n",
    "\\right]+\n",
    "x_4\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "5 \\\\ 4 \\\\ 0\\\\1 \\\\0\n",
    "\\end{matrix}\n",
    "\\right]+\n",
    "x_5\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "-6 \\\\ -4 \\\\ 0\\\\0 \\\\1\n",
    "\\end{matrix}\n",
    "\\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3. Perform rref on augmented $A$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "4. Transpose $A$ then do rref."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & 0 & -2 & 1\\\\0 & 1 & -3 & 2\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right], \\  \\left( 0, \\  1\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  0  -2  1⎤        ⎞\n",
       "⎜⎢           ⎥        ⎟\n",
       "⎜⎢0  1  -3  2⎥        ⎟\n",
       "⎜⎢           ⎥        ⎟\n",
       "⎜⎢0  0  0   0⎥        ⎟\n",
       "⎜⎢           ⎥, (0, 1)⎟\n",
       "⎜⎢0  0  0   0⎥        ⎟\n",
       "⎜⎢           ⎥        ⎟\n",
       "⎜⎢0  0  0   0⎥        ⎟\n",
       "⎜⎢           ⎥        ⎟\n",
       "⎝⎣0  0  0   0⎦        ⎠"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.T.rref()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are 2 pivot columns."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Actually, we don't need any calculation to know the rank of $A^T$, because\n",
    "\n",
    "$$\n",
    "rank(A)=rank(A^T)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <font face=\"gotham\" color=\"purple\"> Orthogonality of $\\text{Nul}A$ and $\\text{Row}A$ </font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##  <font face=\"gotham\" color=\"purple\"> $\\text{Nul}A \\perp \\text{Row}A$  </font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is the intersting connections of these subspaces we have discussed. Consider"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}5 & 8 & 2\\\\10 & 16 & 4\\\\3 & 4 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡5   8   2⎤\n",
       "⎢         ⎥\n",
       "⎢10  16  4⎥\n",
       "⎢         ⎥\n",
       "⎣3   4   1⎦"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = sy.Matrix([[5, 8, 2], [10, 16, 4], [3, 4, 1]]);A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & 0 & 0\\\\0 & 1 & \\frac{1}{4}\\\\0 & 0 & 0\\end{matrix}\\right], \\  \\left( 0, \\  1\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  0   0 ⎤        ⎞\n",
       "⎜⎢         ⎥        ⎟\n",
       "⎜⎢0  1  1/4⎥, (0, 1)⎟\n",
       "⎜⎢         ⎥        ⎟\n",
       "⎝⎣0  0   0 ⎦        ⎠"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.rref()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The basis of row space of $A$ is $(1, 0, 0)$ and $(0, 1, .25)$.And the $\\text{Row}A$ is \n",
    "\n",
    "$$\n",
    "\\text{Row}A=\n",
    "s\\left[\n",
    "\\begin{matrix}\n",
    "1 \\\\ 0\\\\ 0\n",
    "\\end{matrix}\n",
    "\\right]+\n",
    "t\\left[\n",
    "\\begin{matrix}\n",
    "0 \\\\ 1\\\\ 0.25\n",
    "\\end{matrix}\n",
    "\\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The $\\text{Nul}A$ is \n",
    "$$\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "x_1 \\\\ x_2\\\\ x_3\n",
    "\\end{matrix}\n",
    "\\right]=\n",
    "x_3\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "0 \\\\ -.25\\\\ 1\n",
    "\\end{matrix}\n",
    "\\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can visualize their relations geometrically. Again keep in mind that Matplotlib does not render 3D properly, so you need some imagination as well.\n",
    "\n",
    "Here is what we observe. \n",
    "\n",
    "The $\\text{Row}A$ is a plane and $\\text{Nul}A$ is a line which is perpendicular to the plane. It is easy to grasp the idea if you notice that in a homogeneous system $Ab = \\mathbf{0}$, it breaks down into many dot products\n",
    "\n",
    "$$\n",
    "Ab =\\left[\n",
    "\\begin{matrix}\n",
    "A_{1i}\\cdot b \\\\ A_{2i}\\cdot b\\\\ A_{3i}\\cdot b\n",
    "\\end{matrix}\n",
    "\\right]\n",
    "$$\n",
    "\n",
    "where $A_{1i}, A_{2i}, A_{3i}$ are the rows of $A$. In later chapters we will prove when the dot product of two vectors equals zero, which means geometrically they are perpendicular."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "s = np.linspace(-1, 1, 10)\n",
    "t = np.linspace(-1, 1, 10)\n",
    "S, T = np.meshgrid(s, t)\n",
    "\n",
    "X = S\n",
    "Y = T\n",
    "Z = T*.25\n",
    "\n",
    "fig = plt.figure(figsize = (8,8))\n",
    "\n",
    "ax = fig.add_subplot(111,projection='3d')\n",
    "ax.plot_surface(X, Y, Z, alpha = .9, cmap=plt.cm.coolwarm)\n",
    "\n",
    "x3 = np.linspace(-1, 1, 10)\n",
    "x1 = 0*x3\n",
    "x2 = -.25*x3\n",
    "ax.plot(x1,x2,x3, lw = 5)\n",
    "\n",
    "ax.set_xlabel('x-axis', size = 18)\n",
    "ax.set_ylabel('y-axis', size = 18)\n",
    "ax.set_zlabel('z-axis', size = 18)\n",
    "\n",
    "ax.axis([-1,1,-1,1])\n",
    "\n",
    "ax.text(x = 1, y = -1, z = -.25, s = r'$Row\\ A$', size = 17)\n",
    "ax.text(0, -.25, 1, s = r'$Nul\\ A$', size = 17)\n",
    "\n",
    "ax.view_init(7, 20)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##  <font face=\"gotham\" color=\"purple\"> $\\text{Nul}A^T \\perp \\text{Col}A$  </font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The nullity of $A^T$ is"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & 2 & 0\\\\0 & 0 & 1\\\\0 & 0 & 0\\end{matrix}\\right], \\  \\left( 0, \\  2\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  2  0⎤        ⎞\n",
       "⎜⎢       ⎥        ⎟\n",
       "⎜⎢0  0  1⎥, (0, 2)⎟\n",
       "⎜⎢       ⎥        ⎟\n",
       "⎝⎣0  0  0⎦        ⎠"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = sy.Matrix([[5, 8, 2], [10, 16, 4], [3, 4, 1]]);A.T.rref()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The $\\text{Nul}A^T$ is \n",
    "\n",
    "$$\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "x_1 \\\\ x_2\\\\ x_3\n",
    "\\end{matrix}\n",
    "\\right]=\n",
    "x_2\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "-2 \\\\ 1\\\\ 0\n",
    "\\end{matrix}\n",
    "\\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The  $\\text{Col}A$ is "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & 0 & 0\\\\0 & 1 & \\frac{1}{4}\\\\0 & 0 & 0\\end{matrix}\\right], \\  \\left( 0, \\  1\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  0   0 ⎤        ⎞\n",
       "⎜⎢         ⎥        ⎟\n",
       "⎜⎢0  1  1/4⎥, (0, 1)⎟\n",
       "⎜⎢         ⎥        ⎟\n",
       "⎝⎣0  0   0 ⎦        ⎠"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.rref()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\text{Col}A=\n",
    "s\\left[\n",
    "\\begin{matrix}\n",
    "5 \\\\ 10\\\\ 3\n",
    "\\end{matrix}\n",
    "\\right]+\n",
    "t\\left[\n",
    "\\begin{matrix}\n",
    "8 \\\\ 16\\\\ 4\n",
    "\\end{matrix}\n",
    "\\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\text{Col}A$ is a plane and $\\text{Nul}A^T$ is a line perpendicular to the plane. The intuition is similar to $\\text{Nul}A \\perp \\text{Row}A$, here you can think of a system look like $b^TA = \\mathbf{0}^T$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "s = np.linspace(-1, 1, 10)\n",
    "t = np.linspace(-1, 1, 10)\n",
    "S, T = np.meshgrid(s, t)\n",
    "\n",
    "X = 5*S+8*T\n",
    "Y = 10*S+16*T\n",
    "Z = 3*S+4*T\n",
    "\n",
    "fig = plt.figure(figsize = (8,8))\n",
    "\n",
    "ax = fig.add_subplot(111,projection='3d')\n",
    "ax.plot_surface(X, Y, Z, cmap=plt.cm.coolwarm)\n",
    "\n",
    "x2 = np.linspace(-1, 1, 10)\n",
    "x3 = x2*0\n",
    "x1 = -2*x2\n",
    "\n",
    "ax.plot(x1,x2,x3, lw = 3)\n",
    "\n",
    "ax.set_xlabel('x-axis', size = 18)\n",
    "ax.set_ylabel('y-axis', size = 18)\n",
    "ax.set_zlabel('z-axis', size = 18)\n",
    "\n",
    "ax.axis([-1,1,-1,1])\n",
    "\n",
    "ax.view_init(-67, 35)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <font face=\"gotham\" color=\"purple\"> Rank Decomposition </font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider a matrix $A$, the purpose is to decompose it into the multiplication of $C$, $R$, which are the  bases of column space and row space respectively."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "A = CR\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2 & 4 & 1 & -1\\\\4 & 2 & -4 & 2\\\\2 & -2 & -5 & 3\\\\1 & 9 & -3 & 2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡2  4   1   -1⎤\n",
       "⎢             ⎥\n",
       "⎢4  2   -4  2 ⎥\n",
       "⎢             ⎥\n",
       "⎢2  -2  -5  3 ⎥\n",
       "⎢             ⎥\n",
       "⎣1  9   -3  2 ⎦"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = sy.Matrix([[2, 4, 1, -1], [4, 2, -4, 2], [2, -2, -5, 3], [1, 9, -3, 2]]);A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & 0 & 0 & - \\frac{4}{21}\\\\0 & 1 & 0 & \\frac{1}{63}\\\\0 & 0 & 1 & - \\frac{43}{63}\\\\0 & 0 & 0 & 0\\end{matrix}\\right], \\  \\left( 0, \\  1, \\  2\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  0  0  -4/21⎤           ⎞\n",
       "⎜⎢              ⎥           ⎟\n",
       "⎜⎢0  1  0  1/63 ⎥           ⎟\n",
       "⎜⎢              ⎥           ⎟\n",
       "⎜⎢         -43  ⎥, (0, 1, 2)⎟\n",
       "⎜⎢0  0  1  ──── ⎥           ⎟\n",
       "⎜⎢          63  ⎥           ⎟\n",
       "⎜⎢              ⎥           ⎟\n",
       "⎝⎣0  0  0    0  ⎦           ⎠"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Arref = A.rref();Arref"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Get the basis of $\\text{Col}A$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2 & 4 & 1\\\\4 & 2 & -4\\\\2 & -2 & -5\\\\1 & 9 & -3\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡2  4   1 ⎤\n",
       "⎢         ⎥\n",
       "⎢4  2   -4⎥\n",
       "⎢         ⎥\n",
       "⎢2  -2  -5⎥\n",
       "⎢         ⎥\n",
       "⎣1  9   -3⎦"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ColA_basis = A[:,:3];ColA_basis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then get the $\\text{Row}A$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0 & - \\frac{4}{21}\\\\0 & 1 & 0 & \\frac{1}{63}\\\\0 & 0 & 1 & - \\frac{43}{63}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  0  0  -4/21⎤\n",
       "⎢              ⎥\n",
       "⎢0  1  0  1/63 ⎥\n",
       "⎢              ⎥\n",
       "⎢         -43  ⎥\n",
       "⎢0  0  1  ──── ⎥\n",
       "⎣          63  ⎦"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "RowA_basis = Arref[0][0:3,:];RowA_basis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Multiply $CR$, we are getting back $A$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2 & 4 & 1 & -1\\\\4 & 2 & -4 & 2\\\\2 & -2 & -5 & 3\\\\1 & 9 & -3 & 2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡2  4   1   -1⎤\n",
       "⎢             ⎥\n",
       "⎢4  2   -4  2 ⎥\n",
       "⎢             ⎥\n",
       "⎢2  -2  -5  3 ⎥\n",
       "⎢             ⎥\n",
       "⎣1  9   -3  2 ⎦"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ColA_basis*RowA_basis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Verify if $CR$ equals $A$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ColA_basis*RowA_basis == A"
   ]
  }
 ],
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